Bent functions shows some vital properties among all combinatorial
objects. Its links in combinatorics, cryptography and coding theory attract the scientific community to construct new class of bent functions. Since the entire characterisation of bent functions is still unexplored but several construction on different algebraic structure is in progress. In this paper we proposed a generalized Maiorana-McFarland construction of bent function from Galois ring

B. K. Dass

P. R. Mishra

Shashi Kant Pandey

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Noname manuscript No.(will be inserted by the editor)A Maiorana-McFarland Construction of a GBF on GaloisringShashi Kant Pandey · P.R.Mishra· B.K.Dassthe date of receipt and acceptance should be inserted laterAbstract Bent functions shows some vital properties among all combinatorialobjects. Its links in combinatorics, cryptography and coding theory attract thescientific community to construct new class of bent functions. Since the entirecharacterisation of bent functions is still unexplored but several constructionon different algebraic structure is in progress. In this paper we proposed ageneralized Maiorana-McFarland construction of bent function from Galoisring.Keywords: Bent Function, Galois ring, Non Linearity.1 IntroductionRothous [1] was proposed the word Bent for the function having flat Walshtransformation. In this paper he showed various properties of bent functionsover binary field. The importance of these functions in Cryptography and Cod-ing theory attracts the scientific community very much so from beginning totill today these functions are playing important role in designing and analysingof a cryptosystem. Later Kumar, Scholtz and Welch generalized the functionand called it generalized q-arey bent function. Proper characterisation of thesefunctions is still a challenging problem and designing of a bent function givesa lot of clues about inner nature of these functions. Before the construction ofa bent function on a specific finite algebraic structure it is always beneficial todifferentiate the limitation of the existence and non existence of a q-ary func-tion. A lot of construction is available in this direction on various algebraicShashi Kant Pandey · B.K. DassDepartment of Mathematics, University of Delhi, Delhi-110007, India,E-mail: shashikantshvet@gmail.comE-mail: dassbk@rediffmail.comP.R. MishraScientific Analysis Group, Metcalfe House Complex, DRDO, Delhi-110054, IndiaE-mail: prasanna.r.mishra@gmail.com2 Shashi Kant Pandey et al.structures[2]. In parallel study of these functions people are exploring otherfunctions having almost similar properties viz. planner function, APN func-tions, PN functions and almost bent functions. A large literature is availableabout the characterisation of these functions[3].In this paper we are giving more generalized construction of a bent functionbased on a ring structure. Here the ring structure is a Galois ring of typeGR(qk, n), where q, k and n are any positive integers. Further distinguishinga field structure from the Galois ring and proposed a generalized constructionover this field. Kai Schmidth explore several bent construction for CDMAcode in[4]. He proposed a construction based on the GR(2h, 1), where h isany integer. Taking similar practice we proposed a construction which is moregeneral construction from Schmidth.The background of bent function is available in section 2. The character-isation of Galois ring and the proof of leading theorem is available in section3. Section 4 contains the description of construction of bent function.2 Bent FunctionThe word Bent is first time proposed by Rothous for the functions having aspacial spectral property. Let q, p and n be any two positive integers and ζkbe the kth root of unity then the Walsh transformation of a function from Zqnto Zp is defined asWf (w) =∑x∈Zqnζqf(x)ζpw.x, w ∈ Zqn (1)Where w.x = (w1.x1 + w2.x2 + ... + wn.xn) mod p for w, x ∈ Zqn. Wf (w)√qn iscalled normalised Walsh transformation of f . The values of Wf (w) are calledthe Walsh coefficient of f . To measure the distance between f and set of alllinear and affine functions, these coefficients are useful. So maximum distancecan be achievable when we have the optimum minimum value of |Wf (w)|. Afunction f is called bent if the minimum optimum value of |Wf (w)| =√qn forall w ∈ Zqn. A function f̂ from Zqn to Zp is called dual of f if Wf (w) = ζ f̂ . Notalways Bent function exist for all integer values of p, q and n. Which makes itmore interesting and rare among the combinatorial objects. There are otheranalogous functions available like planer function, APN, PN function etc. Forp = q = 2 Bent function always exist for all even values of n and for odd n itmay or may not exist. There are several cases available[5] for p, q and n wherebent functions do not exist.3 Galois RingThe Galois ring is much similar as Galois field but there are some difference be-tween the two algebraic structure. Specially they differ from the multiplicativeA Maiorana-McFarland Construction of a GBF on Galois ring 3operation point of view. The major difference is that the division is not possi-ble in general in a Galois ring whereas all the elements of a Galois field havetheir multiplicative inverse. The availability of multiplicative inverse gives alot of analogous representation of an element in a Galois field. A Galois Ringhave also various representation but many of the representation are remainunexplored.Let p be a prime number n,m be any positive integers. The Galois ringG(pn,m) is the extension of the ring Zpn of degree m. If g is a monic basicirreducible polynomial over Znp of degree m then GR(pn,m) can be seen asZpn [X]/ < g > having pmn elements. Hence GR(p,m) is equivalent to theGalois field of order pm. Let η be the primitive element in GR(pn,m) of orderpm − 1 and Tn = {0, 1, η, η2, ..., ηpm−2} is the Teichmuller set. Every elementof GR(pn,m) can be represented in two different way, for any z ∈ GR(pn,m)The multiplicative form:z =n∑i=1pi−1ti, ti ∈ Tn (2)The additive form:z =n∑i=0aiηi, ai ∈ Zpn (3)The set of Teichmuller representatives in GR(pn, 1) can also be seen asF = {z ∈ Zpn : zpn= z}So the multiplicative form of an element of GR(pn, 1) is basically its uniquelywritten p-adic expansion. Now we define two operations on F by a ⊕ b =(a+ b)pn−1and a⊗ b = a.b mod pn for all a, b ∈ F .Lemma 1 Let p be any prime and z be any integer, if p - z then pn - z.Lemma 2 Let p be any prime and z be any integer, if p | z then p - zpn−1−1.Theorem 31. For any prime p,(F,⊕,⊗) is a field and |F | = p.Proof It is clear that 0 ∈ F . Now let any 0 6= z ∈ F . Thenz(zpn−1 − 1) = 0 (4)Case1. If p - z:By Lemma1., pn and z are relatively prime. Hence there exist two integers a, bsuch thatapn + bz = 1 (5)Taking modulo with pn both side we get the multiplicative inverse of z in F .Case2. If p | z:4 Shashi Kant Pandey et al.By Lemma2. and above argument zpn−1−1 has multiplicative inverse. There-fore by the argument of equation 4 and 5, z = 0. Hence in any case 0 6= z ∈ F ,then z is a unit in Zpn . Note that units in Zpn form a group. Moreover ,if Z?pndenote the group of units in Zpn thenZ?pn∼= Cp−1 ⊕ Cpn−1 . (6)Where Cn denotes the cyclic group of order n and ⊕ is the direct sum. Again,if 0 6= z ∈ F , then z is a unit and hence by equation 4, zpn−1 = 1. So the mul-tiplicative order of z should also divides pn − 1. By equation 6 ,multiplicativeorder of z should also divides (p−1)(pn−1). Which implies that multiplicativeorder of z divides p− 1. Hencez ∈ Zp−1 ⊆ Cp−1 ⊕ Cpn−1Or via any isomorphism φ on Z?pnφ(F − {0}) ⊆ Zp−1Now conversely suppose that z ∈ Cp−1 ⊆ Cp−1 ⊕ Zpn−1 . Note that Zp−1 isa subgroup of an isomorphic image of units of Zpn . Hence zp−1 = 1. Thuszp = z =⇒ zp2 = zp = z. Inductively zpn = z =⇒ Zp−1 ⊆ F viaisomorphism. Thus we have shown that F − {0} ∼= Zp−1. Hence |F | = p as0 ∈ F .The operation ⊗ is same as that of Zpn so obviously it is closed inderthis. From case1. and 2. it is clear that every non zero element of F havingmultiplicative inverse in F .Net, let a, b ∈ F . Then apn = a and bpn = b. If any of a and b is 0 thena⊕ b ∈ F is trivial. So assume that a 6= 0 and b 6= 0 so ap−1 = bp−1 = 1 =⇒ap = a and bp = b. Also note that,(a+ b)p = ap + bp( mod p) = a+ b( mod p)=⇒ (a+ b)p2= (a+ b)p( mod p) = a+ b( mod p)Inductively,(a+ b)pn= a+ b( mod p)=⇒ ((a+ b)pn)pn−1= (a+ b)pn−1( mod pn)i.e(a⊕ b)pn= a⊕ bHence a⊕ b ∈ F . Now to show distribution of ⊗ over ⊕, Let c ∈ F then,c⊗ a⊕ c⊗ b = (c⊗ a+ c⊗ b)pn= cpn⊗ (a+ b)pn= c⊗ (a⊕ b)Thus (F,⊕,⊗) is a field of order p.A Maiorana-McFarland Construction of a GBF on Galois ring 54 Construction of bent function from Galois RingOn the basis of selection space of input variables there are other generaliza-tion of boolean function is available in litrature [6]. Here we are showing ageneralized construction of Bent function from Galois rings.Theorem 41. Let π be a permutation on F k, g be a function from F k to Zph ,h and k are any two integers and the function f : F 2k 7→ Zph is defined asf(x, y) = g(x)− ph−1π(x).y.Then f is a bent function.Proof The coefficient walsh transform of f can be written asWf (u, v) =∑x,y∈Fkζg(x)−ph−1π(x).yphζu.x+v.yp=∑x∈Fkζg(x)phζu.xp∑y∈Fk(ζp)(v−π(x)).yIf π(x) 6= v then the inner sum will be zero and if π(x) = v then the innersum gives the value pk. So we can write the walsh coefficient should beWf (u, v) = pkζg(π−1(v))phζu.π−1(v)pHence f is bent.References1. O. S. Rothaus, On Bent functions, J.Comb.Theory(A) 20(1976),300-305.2. Claude Carlet, A construction of Bent function,London mathematical SocietyLecture Note Series 233.3. Celine Blondeau ,Kaisa Nyberg, Perfect nonlinear function and cryptography,Finite Field and Their Applications 32(2015),120-147.4. Kai-Uwe Schmidt, Quarternary Constant-Amplitude Codes for MulticodeCDMA,arXiv:cs/0611162v2.5. P.V Kumar, R.A Scholtz, L.R Welch,Generalized bent functions and their prop-erties,Journal of Combinatorial Theory, Series A Volume 40, Issue 1,1985,90-107.6. N.N.Tokareva, Generalizations of bent functions. A survey,Journal of Appliedand Industrial Mathematics, Volume 5, Issue 1,2011,110-129.7. q-ary Bent Functions Constructed from Chain Rings,Xiang-dong Hou,FINITEFIELDS AND THEIR APPLICATIONS4,5561 (1998).8. Bjorn Abrahamsson, Architectures for Multiplication in GaloisRings,http://www.ep.liu.se/Linkping University Electronic Press.9. Javier Gomez-Calderon, Gary L. Mullen, Galois ring and algebraic cryptogra-phy, Acta Arithmetica LIX4.1991,317-328.

A Maiorana-McFarland Construction of a GBF on Galois ring

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